For \(k=0,\dots ,m\) , all derivatives \({\frac{\mathrm {d}^{k}}{\mathrm {d}t^{k}}}\widetilde{\varphi }_{T_n^{\prime }}\)  exist and are continuous. Furthermore, \({\frac{\mathrm {d}^{k}}{\mathrm {d}t^{k}}}\widetilde{\varphi }_{T_n^{\prime }}\)  is bounded by \(c(m)n^k|t|^k\exp (-\beta _nt^2/2)\in L^1\)  and thus in \(L^1\) . The Fourier transform of our approximating function \(\widetilde{\varphi }_{\Phi ^{\widetilde{P}}_{m,n}}(t)\)  also possesses these properties by eq.ddtl-tphi. This means that the conditions of the Fourier inversion theorem (see e.g. [1]}) are satisfied. This yields
\(\sup _{x\in \mathbb {R}} |x|^{k}|\widetilde{f}_n^{\prime }(x)-\phi ^{\widetilde{p}}_{m,n}(x)|\le \int _{-\infty }^{\infty } \Big |{\frac{\mathrm {d}^{k}}{\mathrm {d}t^{k}}}\big (\widetilde{\varphi }_{T_n^{\prime }}(t)-\widetilde{\varphi }_{\Phi ^{\widetilde{P}}_{m,n}}(t)\big )\Big |\,\mathrm {d}t\) 
